Welcome Susan!
It appears to be Lavandula multifida (Fernleaf Lavender.) It should be hardy to about zone 8, if drainage is good. It sounds like you may want to take it with you to zone 8b. If you give it a decent size pot, you should be able to let it stay outside for most of the winter there and only bring it in if you get a really hard freeze. It's a Mediterranean plant, and they usually don't do well as long-term house plants. See link:
Fernleaf Lavender (Lavandula multifida)
Best of luck!
